The importance of precise homography estimation is often underestimated even though it plays a crucial role in various vision applications such as plane or planarity detection, scene degeneracy tests, camera motion classification, image stitching, and many more. Ignoring the radial distortion component in homography estimation-even for classical perspective cameras-may lead to significant errors or totally wrong estimates. In this paper, we fill the gap among the homography estimation methods by presenting two algorithms for estimating homography between two cameras with different radial distortions. Both algorithms can handle planar scenes as well as scenes where the relative motion between the cameras is a pure rotation. The first algorithm uses the minimal number of five image point correspondences and solves a nonlinear system of polynomial equations using Gröbner basis method. The second algorithm uses a non-minimal number of six image point correspondences and leads to a simple system of two quadratic equations in two unknowns and one system of six linear equations. The proposed algorithms are fast, stable, and can be efficiently used inside a RANSAC loop.

The estimation of the epipolar geometry of two cameras from image matches is a fundamental problem of computer vision with many applications. While the closely related problem of estimating relative pose of two different uncalibrated cameras with radial distortion is of particular importance, none of the previously published methods is suitable for practical applications. These solutions are either numerically unstable, sensitive to noise, based on a large number of point correspondences, or simply too slow for real-time applications. In this paper, we present a new efficient solution to this problem that uses 10 image correspondences. By manipulating ten input polynomial equations, we derive a degree 10 polynomial equation in one variable. The solutions to this equation are efficiently found using the Sturm sequences method. In the experiments, we show that the proposed solution is stable, noise resistant, and fast, and as such efficiently usable in a practical Structure-from-Motion pipeline.

In this paper we provide a new fast and stable algebraic solution to the problem of L2triangulation from three views. We use Lagrange multipliers to formulate the search for the minima of the L2objective function subject to equality constraints. Interestingly, we show that by relaxing the triangulation such that we do not require a single point in 3D, we get, after a linear correction, a solver that is faster, more stable and practically as accurate as the state-of-the-art L2-optimal algebraic solvers [24, 7, 8, 9]. In our formulation, we obtain a system of eight polynomial equations in eight unknowns, which we solve using the Groebner basis method. We get less (31) solutions than was the number (47-66) of solutions obtained in [24, 7, 8, 9] and our solver is more robust than [8, 9] w.r.t. critical configurations. We evaluate the precision and speed of our solver on both synthetic and real datasets.

The problem of determining the absolute position and orientation of a camera from a set of 2D-to-3D point correspondences is one of the most important problems in computer vision with a broad range of applications. In this paper we present a new solution to the absolute pose problem for camera with unknown radial distortion and unknown focal length from five 2D-to-3D point correspondences. Our new solver is numerically more stable, more accurate, and significantly faster than the existing state-of-the-art minimal four point absolute pose solvers for this problem. Moreover, our solver results in less solutions and can handle larger radial distortions. The new solver is straightforward and uses only simple concepts from linear algebra. Therefore it is simpler than the state-of-the-art Groebner basis solvers. We compare our new solver with the existing state-of-the-art solvers and show its usefulness on synthetic and real datasets.

In this paper we propose methods for speeding up minimal solvers based on Gröbner bases and action matrix eigenvalue computations. Almost all existing Gröbner basis solvers spend most time in the eigenvalue computation. We present two methods which speed up this phase of Gröbner basis solvers: (1) a method based on a modified FGLM algorithm for transforming Gröbner bases which results in a single-variable polynomial followed by direct calculation of its roots using Sturm-sequences and, for larger problems, (2) fast calculation of the characteristic polynomial of an action matrix, again solved using Sturm-sequences. We enhanced the FGLM method by replacing time consuming polynomial division performed in standard FGLM algorithm with efficient matrix-vector multiplication and we show how this method is related to the characteristic polynomial method. Our approaches allow computing roots only in some feasible interval and in desired precision. Proposed methods can significantly speedup many existing solvers. We demonstrate them on three important minimal computer vision problems.

We present a method for solving systems of polynomial equations appearing in computer vision. This method is based on polynomial eigenvalue solvers and is more straightforward and easier to implement than the state-of-the-art Gröbner basis method since eigenvalue problems are well studied, easy to understand, and efficient and robust algorithms for solving these problems are available. We provide a characterization of problems that can be efficiently solved as polynomial eigenvalue problems (PEPs) and present a resultant-based method for transforming a system of polynomial equations to a polynomial eigenvalue problem. We propose techniques that can be used to reduce the size of the computed polynomial eigenvalue problems. To show the applicability of the proposed polynomial eigenvalue method, we present the polynomial eigenvalue solutions to several important minimal relative pose problems.

In this paper we aim at reconstructing 3D scenes from images with unknown focal lengths downloaded from photosharing websites such as Flickr. First we provide a minimal solution to finding the relative pose between a completely calibrated camera and a camera with an unknown focal length given six point correspondences. We show that this problem has up to nine solutions in general and present two efficient solvers to the problem. They are based on Gröbner basis, resp. on generalized eigenvalues, computation. We demonstrate by experiments with synthetic and real data that both solvers are correct, fast, numerically stable and work well even in some situations when the classical 6-point algorithm fails, e.g. when optical axes of the cameras are parallel or intersecting. Based on this solution we present a new efficient method for large-scale structure from motion from unordered data sets downloaded from the Internet. We show that this method can be effectively used to reconstruct 3D scenes from collection of images with very few (in principle single) images with known focal lengths.

In this paper we provide new fast and simple solutions to two important minimal problems in computer vision, the five-point relative pose problem and the six-point focal length problem. We show that these two problems can easily be formulated as polynomial eigenvalue problems of degree three and two and solved using standard efficient numerical algorithms. Our solutions are somewhat more stable than state-of-the-art solutions by Nister and Stewenius and are in some sense more straightforward and easier to implement since polynomial eigenvalue problems are well studied with many efficient and robust algorithms available. The quality of the solvers is demonstrated in experiments 1. 1

This paper presents a general solution to the determination of the pose of a perspective camera with unknown focal length from images of four 3D reference points. Our problem is a generalization of the P3P and P4P problems previously developed for fully calibrated cameras. Given four 2D-to-3D correspondences, we estimate camera position, orientation and recover the camera focal length. We formulate the problem and provide a minimal solution from four points by solving a system of algebraic equations. We compare the Hidden variable resultant and Grobner basis techniques for solving the algebraic equations of our problem. By evaluating them on synthetic and on real-data, we show that the Grobner basis technique provides stable results.

We pose robust matching with parametric and non-parametric constraints as the problem of finding a stable independent set (SIS) in an oriented graph whose vertices are all possible correspondences, whose edges capture the structure of the constraints and whose edge orientation represents pairwise comparison 'is better' based on correspondence quality, including the uncertainty of this comparison. We show SIS possess properties of both robustness and weak optimality. The main contribution of this paper is algorithmic speedup that results from exploiting the dependence between the standard uniqueness constraint and the parametric constraint. The general theory is demonstrated on the example of image stitching using homography model. The algorithm needs at most 𝜅N2calls of a procedure testing if two ellipse correspondences are consistent with a general homography. The previous known SIS algorithm needed O(N4) tests. Experiments show the method gives good results and is fast in practice with 𝜅≈0.3.